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| | btq.tex |
 | | With respect to the induced metric $h^{(m)}$ on $L^m$ we obtain now the corresponding scalar products on the space of global sections of $L^m$, the projection operator \begin{equation} \Pi^{(m)}:\gulm\to\ghm, \end{equation} the Toeplitz operators \begin{equation} \Tfm:\ghm\to\ghm, \end{equation} and the Berezin-Toeplitz quantization map \begin{equation} T^{(m)}:\cim\to\eghm\, \end{equation} for every $m\in\N$. |
| | I will show for compact K\"ahler manifolds that the Toeplitz operator map and the covariant symbol map of Berezin are adjoint if one takes the Hilbert-Schmidt norm for the operators and the, by the epsilon function of Rawnsley, deformed Lebesgue measure for the functions. |
| | for $\hbar=\frac 1m\to 0$) as shown by Bordemann, Meinrenken and Schlichenmaier \cite{BMS}: \begin{theorem}\label{T:toeapp} For $f,g\in\cim$ we have \begin{alignat}{2} \mathrm{(a)}\qquad& \qquad\qquad\qquad&\Tfm\qquad\tof_\infty,\qquad &m\to\infty,\\ \mathrm{(b)}\qquad&\qquad&m\;\i[\Tfm,\Tfm[g]]-\Tfm[\{f,g\}]\to \quad0 \quad,\qquad &m\to\infty,\\ \mathrm{(c)}\qquad&\qquad&\Tfm \Tfm[g]-\Tfm[f\cdot g]\quad\to \quad0\quad,\qquad &m\to\infty, \end{alignat} where for the operators the operator norm and for the functions the sup-norm have been chosen. |
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